Alexia, Pavlos, Nikolaos, Theodoran and Phillipos didn't inherited the title prince(ss) of Denmark from there mother. 'HM (elected) King George I of Greece (Hellenes)' was born 'HRH Prince George of Denmark', the second son of 'HM King Christian IX of Denmark', making his male line descendents prince(ss) of Denmark

If you consider this, I don't think that the Children of 'HM Queen Sofia of Spain, Princess Of Greece and Denmark' bear the title: Prince(ss) of Greece and Denmark

If I'm correct, the first king of Greece (George I) was elected to throne. Prior to that, he was simply His Royal Highness Prince William of Denmark. He didn't adopt the name "George" until he ascended the Greek (or more formally, Hellenic) throne. As he was a prince of Denmark, his descent in the male line carry the title Prince(ss) of Greece and Denmark.

I'm not totally sure, but it comes from the marriage of Queen Juana "la Loca" of Castile to Archduke Philip of Austria; in this way the title became one of the titles of the King of Spain, and thus has been inherited by King Felipe V (although he was a Bourbon) and therefore now it is one of the titles of the present King.

I'm not totally sure, but it comes from the marriage of Queen Juana "la Loca" of Castile to Archduke Philip of Austria; in this way the title became one of the titles of the King of Spain, and thus has been inherited by King Felipe V (although he was a Bourbon) and therefore now it is one of the titles of the present King.

Felipe V was a Bourbon, but his grandmother the Queen of France was a Hasburg and was born a Spanish Infanta. She was half-sister of the last Hasburg Spanish King Carlos II. Maybe they used that angle to inherit all the titles associated to the King of Spain?

I don't think, for a simple reason: the title is indeed associated to the Crown of Spain since the time of the Habsburgs reign (before 1700), and this is the reason because Juan Carlos still bears it; but that title can't be transmitted from a woman to her children, it can be transmitted only from an Archduke to his children, with the due exceptions (in the case of a total extinction of male Archdukes, and in this case the closest Archduchess by birth to the last male becomes Head of the Austrian Imperial and Archducal Family and thus can transmit her titles to her children; i.e. Maria Theresia, Holy Roman Emperor Charles VI's daughter).
Maria Teresa (Louis XIV's wife and grandmother of Felipe V) didn't lose her rights to the Spanish Throne, as her dowry wasn't fully paid to the French King, but this doesn't imply that she could transmit her Austrian titles to her offspring.

I believe when the claims for these titles were first made (when ever that was for that particular title) there was merit under whatever succession law the king claimed them under. Since then, other legal processes and conventions have since rendered many of the titles 'in pretense'.

In English common law there is a saying... 'possession is nine-tenths of of the law.'

I understand that Iñaki Urdangarín is styled as His Excellency the Duke of Palma de Mallorca because he is married to HRH Infanta Cristina the Duchess of Parma de Mallorca. In Spain, they seem to do these things differently. Infanta Cristina's elder sister Infanta Elena was created as the Duchess of Lugo upon her marriage to Don Jaime de Marichalar y Sáenz de Tejada who is now styled as HE the Duke (consort) of Lugo.

Exactly. In Spain titles are shared by the other spouse. Because the Infantas Elena and Cristina were created Duchess of X and Y, her husbands are(or were, in the case of Jaime de Marichalar) known as Duke of X and Y, but they are only the consorts.

I wonder what would have happened if the Infantas had decided against marriage, eg like their aunt Irene. Both sisters only got their titles upon marriage, not earlier, they probably would have remained Infantas of Spain (not bad either).

Who knows, maybe the King would had granted them a dukedom when they reached a certain age. In any case, unlike Britain for example, where a prince or a princess can legally be a commoner when they are not peers, the infanta title is much more important than the dukedom, maybe only granted so that the consort enjoys a title. When Infanta Cristina signs the Christmas cards, she always signs as Infanta of Spain, never as Duchess of Palma de Mallorca (Iñaki Urdangarín signs as Duke of Palma de Mallorca).